Equation of Motion

Equation (7.46) contains the velocity~v_c of the vortex core in a cross product and a scalar product.

We can transform this equation to an equation of motion that is explicit in ~v_c. By multiplying equation (7.46) with−D_Γα we get

−D_Γα ~F −D_Γα ~G×~v_c−D_Γα ~G×b_j~j−D_Γ²α²~v_c−D_ΓαD₀ξb_j~j = 0. (7.47) Taking the cross product of equation (7.46) withG~ leads to

G~ ×F~ −G²₀~v_c−G²₀b_j~j+G~ ×D_Γα~v_c+G~ ×D₀ξb_j~j = 0. (7.48)

Here we used equation (1.14) and the orthogonality of G~ and~vc. The sum of equation (7.47) and equation (7.48)

(G²₀+D_Γ²α²)~vc =−DΓα ~F +G~ ×F~ −DΓαbjG~ ×~j+D0ξbjG~ ×~j−DΓαD0ξbj~j−G²₀bj~j (7.49) leads to the sought equation

~v_c = −D_Γα ~F +G₀~e_z×F~ −D_Γαb_jG₀~e_z×~j+D₀ξb_jG₀~e_z×~j−D_ΓαD₀ξb_j~j−G²₀b_j~j

G²₀+D²_Γα² (7.50)

for the velocity~vc of the vortex core. The force F~ =−d(E_s+E_z)

d ~R =−µ₀M_sHldh

sinπc 2

~e_x+ncosπc 2

~e_yi

−mω_r²X~e_x−mω²_rY ~e_y (7.51) can be calculated from the energies in equations (7.36) and (7.41). With this we can also calculate the expression

~e_z×F~ =−µ₀M_sHldh

sinπc 2

~e_y−ncosπc 2

~e_xi

−mω_r²X~e_y +mω_r²Y ~e_x. (7.52) We restrict ourselves to a current in x direction that is given by

~j =j~e_x (7.53)

and its cross product with the z direction is given by

~e_z×~j=j~e_y. (7.54)

Inserting the force and the current in equation (7.50) we get X˙

Y˙

=− mω²_r G²₀+D²_Γα²

−D_ΓαX−G₀Y

−D_ΓαY +G₀X

− µ₀M_sHld G²₀+D_Γ²α²

−D_Γαsin ^πc₂

−G₀ncos ^πc₂

−D_Γαncos ^πc₂

+G₀sin ^πc₂

− G0bjj G²₀+D²_Γα²

G₀ D_Γα

− D0ξbjj G²₀+D²_Γα²

D_Γα

−G₀

.

(7.55)

Here and hereafter we discard thez component since the core does not move in this direction.

In the absence of current and field the equation becomes homogeneous. The excited core performs an exponentially damped spiral rotation around its equilibrium position. This rotation is given by

X Y

=C₁ i

np

e^−Γt+iωt+C₂ −i

np

e^{−Γt−iωt} (7.56)

with the free frequency

ω=− npG₀mω_r²

G²₀+D_Γ²α² (7.57)

and the damping constant

Γ =− D_Γαmω_r²

G²₀+D²_Γα². (7.58)

C1 and C2 are constants that have to be chosen so that the displacement for t = 0 fulfills the starting conditions.

With the aid of the free frequency and the damping constant the equation of motion can be written as

X˙ Y˙

=

−Γ −npω npω −Γ

X Y

−µ₀M_sHldnpωG⁻₀¹ ω²+ Γ²

−Γ sin ^πc₂

−pωcos ^πc₂

−Γncos ^πc₂

+npωsin ^πc₂

− npωbjj ω²+ Γ²

npω Γ

−D0npωG⁻₀¹ξbjj ω²+ Γ²

Γ

−npω

.

(7.59)

-15 -10 -5 0 5 10 15

0 10 20 30 40 50

X (nm)

t (ns)

-15 -10 -5 0 5 10 15

0 10 20 30 40 50

Y (nm)

t (ns)

-15 -10 -5 0 5 10 15

-15 -10 -5 0 5 10 15

Y (nm)

X (nm)

Figure 7.2: Free trajectory of the core of a vortex or an antivortex, as described by equation (7.56), with the free frequency ω = 1.03 GHz and the damping constant Γ = 20 MHz. The core starts at X(0) =Y(0) = 10 nm and performs an exponen-tially damped free gyration around its equilibrium position in the center of the sample. The sense of the gyration depends on the product of the wind-ing number n and the polarization p. Shown are the trajectories for np = +1 (solid red line) and np=−1 (dashed blue line).

By rearranging the terms we get X˙

Y˙

=

−Γ −npω npω −Γ

X Y

+µ₀M_sHldpωG⁻₀¹cos ^πc₂ ω²+ Γ²

npω Γ

− npωb_jj ω²+ Γ²

npω Γ

+µ₀M_sHldnpωG⁻₀¹sin ^πc₂ ω²+ Γ²

Γ

−npω

−D₀npωG⁻₀¹ξb_jj ω²+ Γ²

Γ

−npω

.

(7.60)

Inserting the value of the gyrovector that is given in equation (7.26) yields X˙

Y˙

=

−Γ −npω npω −Γ

X Y

−

b_jj+γHlp

2π cosπc 2

^ω2

ω²+Γ² npΓω ω²+Γ²

!

−

D0

G₀

ξbjjnp+γHlnp

2π sinπc 2

^npΓω

ω²+Γ²

−_ω^2ω_+Γ^{2 2}

! .

(7.61)

Harmonic Excitations

In the following we will focus on harmonic excitations, that are j =j₀e^iΩt and H =H₀e^iΩt. The current and the magnetic field are assumed to be in phase. We introduce two abbreviations. In the following we express the current as

˜j_x=b_jj₀ (7.62)

and the field as

H˜_y = γH₀l

2π . (7.63)

Both values have the dimension of a velocity and allow us to compare the field and the current.

The equation of motion is then given by X˙

Y˙

=

−Γ −npω npω −Γ

X Y

−





˜j_x+ ˜H_ypcos ^πc₂ ^DG⁰0

ξ˜j_xnp+ ˜H_ynpsin ^πc₂

− ^DG⁰0

ξ˜j_xnp−H˜_ynpsin ^πc₂ ˜j_x+ ˜H_ypcos ^πc₂





ω² ω²+Γ²

npΓω ω²+Γ²

! e^iΩt.

(7.64)

In this equation both matrices commute. In the trivial case where ˜j_x + ˜H_ypcos (πc/2) = 0 and

|D₀/G₀|ξ˜j_xnp+ ˜H_ynpsin (πc/2) = 0 there is no dynamics of the vortex or antivortex. In all other cases the matrix





˜j_x+ ˜H_ypcos ^πc₂ ^DG⁰0

ξ˜j_xnp+ ˜H_ynpsin ^πc₂

− ^DG⁰0

ξ˜j_xnp−H˜_ynpsin ^πc₂ ˜j_x+ ˜H_ypcos ^πc₂



 (7.65)

is invertible. For these cases we can introduce a new coordinate system ( ˜X,Y˜) that is given by X

Y

=





˜j_x+ ˜H_ypcos ^πc₂ ^DG⁰0

ξ˜j_xnp+ ˜H_ynpsin ^πc₂

− ^DG⁰0

ξ˜j_xnp−H˜_ynpsin ^πc₂ ˜j_x+ ˜H_ypcos ^πc₂



 X˜

Y˜

. (7.66)

Both coordinate systems can be transformed into each other by a rotation and a subsequent scaling.

By inserting equation (7.66) in equation (7.64) and a multiplication with the inverse matrix of equation (7.65) one finds that in the new coordinate system the equation of motion is given by

X˙˜

Y˙˜

!

=

−Γ −npω npω −Γ

X˜ Y˜

−

ω² ω²+Γ²

npΓω ω²+Γ²

!

e^iΩt. (7.67)

This equation can also be written as X˙˜

Y˙˜

!

=

−Γ −npω npω −Γ

X˜ Y˜ +_ω^npω2+Γ²e^iΩt

. (7.68)

Thus in the new coordinate system the equilibrium position of the vortex or antivortex is shifted along the ˜Y axis.

Equation (7.67) is of the form X˙˜

Y˙˜

!

=

−Γ −npω npω −Γ

X˜ Y˜

+

A B

e^iΩt. (7.69)

The solution of this equation is calculated in appendix B and is given by X˜

Y˜

= e^iΩt

ω²+ (iΩ + Γ)²

A(iΩ + Γ)−Bnpω Anpω+B(iΩ + Γ)

. (7.70)

This yields the solution X˜

Y˜

=− e^iΩt ω²+ (iΩ + Γ)²

ω²(iΩ+Γ)

ω²+Γ² −_ω^Γω2_+Γ²²

ω³np

ω²+Γ² +^{npΓω(iΩ+Γ)}_ω2+Γ²

!

=− e^iΩt ω²+ (iΩ + Γ)²

ω²iΩ ω²+Γ²

ωnp+^npΓωiΩ_ω2+Γ²

!

(7.71)

-40 -20 0 20 40

0 2 4 6 8 10

X (nm)

t (ns)

-40 -20 0 20 40

0 2 4 6 8 10

Y (nm)

t (ns)

-40 -20 0 20 40

-40 -20 0 20 40

Y (nm)

X (nm)

Figure 7.3: Trajectory of the core of a vortex or an antivortex that is excited by an alternat-ing spin-polarized current in x direction at res-onance. The dynamics is described by equa-tion (7.72) with ˜j = 1.8 m/s, the free fre-quency ω = 1.03 GHz, the damping constant Γ = 20 MHz, the non-adiabaticity parameter ξ = 0.01, and |D₀/G₀| = 2.26. The sense of the gyration depends on the product of the wind-ing number n and the polarization p. Shown are the trajectories for np = +1 (solid red line) and np=−1 (dashed blue line).

-40 -20 0 20 40

0 2 4 6 8 10

X (nm)

t (ns)

-40 -20 0 20 40

0 2 4 6 8 10

Y (nm)

t (ns)

-40 -20 0 20 40

-40 -20 0 20 40

Y (nm)

X (nm)

Figure 7.4: Trajectory of the core of a vortex or an antivortex that is excited by an alternating magnetic field in y direction at resonance. The chirality is c = 1. The dynamics is described by equation (7.72) with ˜H = 1.8 m/s, the free frequency ω = 1.03 GHz, and the damping con-stant Γ = 20 MHz. The sense of the gyration depends on the product of the winding numbern and the polarizationp. Shown are the trajectories fornp= +1 (solid red line) andnp=−1 (dashed blue line)

that can be transformed back to the original coordinate system to get the sought solution X

Y

=− e^iΩt ω²+ (iΩ + Γ)²





˜j_x+ ˜H_ypcos ^πc₂ ^DG⁰0

ξ˜j_xnp+ ˜H_ynpsin ^πc₂

− ^DG0⁰

ξ˜j_xnp−H˜_ynpsin ^πc₂ ˜j_x+ ˜H_ypcos ^πc₂





ω²iΩ ω²+Γ²

ωnp+^npΓωiΩ_ω2+Γ²

! .

(7.72)

Examples for the dynamics of a harmonically-excited vortex or antivortex are shown in figures 7.3 (current) and 7.4 (field). The solution for a current inydirection and a magnetic field inxdirection can be found from a rotation of the coordinate system by −π/2, that is a rotation of the above solution by π/2, and the replacements of ˜j_x →˜j_y and ˜H_y → −H˜_x. For the case of the antivortex it is important to keep in mind that the rotation also affects the chirality. The chirality can be derived from equation (4.1) that yields c= (φ−nβ)2/π. A rotation by −π/2 increases the angles φ andβ by π/2. This leads to

c^′ = (φ^′−nβ^′)2 π =

φ+π

2 −nβ−nπ 2

2

π = (φ−nβ)2

π + (1−n). (7.73) In general the rotation by −π/2 increases the chirality by 1−n. This leads to the additional replacementc→c−1 +nso that after the rotation the system assumes the desired chirality. The result then reads

X Y

=− e^iΩt ω²+ (iΩ + Γ)²



 ^DG0⁰

ξ˜j_ynp−H˜_xpsin ^πc₂

−˜j_y+ ˜H_xnpcos ^πc₂

˜j_y−H˜_xnpcos ^πc₂ ^D_G0⁰

ξ˜j_ynp−H˜_xpsin ^πc₂





ω²iΩ ω²+Γ²

ωnp+^npΓωiΩ_ω2+Γ²

! .

(7.74)

With equations (7.56), (7.72), and (7.74) we have an analytical description of the dynamics of magnetic vortices and antivortices under harmonic excitations with currents and fields. After the transient states have been damped out the core of the vortex or antivortex gyrates on an elliptical trajectory around its equilibrium position. The sense of gyration depends only on the product of the core polarization p and the winding numbern. For pn= 1 the core gyrates counter clockwise and for pn = −1 the core gyrates clockwise. The field-driven dynamics depend on the chirality while for the current-driven dynamics the trajectory is independent of the chirality.

Pulsed Excitations

In the following we will discuss the excitation of a vortex or an antivortex with a current or field pulse of the form~j(t) = Θ(t−t₀)~j0 orH(t) = Θ(t~ −t₀)H~₀, respectively. Here Θ(t) is the Heaviside step function that is given by

Θ(t) =

0, t≤0

1, t >0. (7.75)

Thus there is no field or current prior to time t=t₀. After this time the current and field have a constant value. A long time after the current or field has been changed, that is all transient states are damped out, the position of the core can be determined from equations (7.72) and (7.74) by setting Ω = 0. This position is given by

X_∞ Y_∞

=− ω

ω²+ Γ²



 ^DG0⁰

ξ˜j_x−˜j_ynp+ ˜H_xcos ^πc₂

+ ˜H_ysin ^πc₂

˜jxnp+ ^DG0⁰

ξ˜jy −H˜xnsin ^πc₂

+ ˜Hyncos ^πc₂



. (7.76)

-30 -15 0 15 30

0 10 20 30 40 50

X (nm)

t (ns)

-30 -15 0 15 30

0 10 20 30 40 50

Y (nm)

t (ns)

-30 -15 0 15 30

-30 -15 0 15 30

Y (nm)

X (nm)

Figure 7.5: Trajectory of the core of a vortex or an antivortex that is excited by spin-polarized cur-rent in x direction. For t <0 the current is zero and the core is at its equilibrium position at the middle of the sample. For t > 0 a constant cur-rent of ˜j = 18 m/s is applied and the core starts to gyrate around a new equilibrium position. The dynamics is described by equation (7.79) with the free frequency ω = 1.03 GHz, the damping con-stant Γ = 20 MHz, the non-adiabaticity parame-ter ξ = 0.01, and|D0/G0|= 2.26. The new equi-librium position depends on the product of the winding numbernand the polarizationp. Shown are the trajectories for np = +1 (solid red line) and np=−1 (dashed blue line).

-30 -15 0 15 30

0 10 20 30 40 50

X (nm)

t (ns)

-30 -15 0 15 30

0 10 20 30 40 50

Y (nm)

t (ns)

-30 -15 0 15 30

-30 -15 0 15 30

Y (nm)

X (nm)

Figure 7.6: Trajectory of the core of a vortex or an antivortex that is excited with the same spin-polarized current as in figure 7.5 but the cur-rent is switched off aftert= 10 ns. Shown is the trajectory before (solid red line) and after (dashed blue line) the current is switched off. During the current pulse the core gyrates around an equilib-rium position that is shifted from the center of the sample. After the current is switched off the core moves back to the center on a spiral trajectory.

Here, only the case np= +1 is shown.

With the aid of equation (7.56) the position including the transient states is written as X

Y

=C₁ i

np

e^{−Γ(t−t0)+iω(t−t0)}+C₂ −i

np

e^{−Γ(t−t0)−iω(t−t0)}+ X_∞

Y_∞

(7.77) where C₁ and C₂ are given by

X₀ Y₀

=C₁ i

np

+C₂ −i

np

+ X_∞

Y_∞

. (7.78)

Here (X0, Y0) is the core position at time t=t0. The solution is given by X

Y

= np(Y₀−Y_∞)−i(X₀−X_∞) 2

i np

e^{−Γ(t−t0)+iω(t−t0)}

+ np(Y₀−Y_∞) +i(X₀−X_∞) 2

−i np

e^{−Γ(t−t0)−iω(t−t0)}+ X_∞

Y_∞

.

(7.79)

For a current-driven excitation an example for a trajectory can be found in figure 7.5. Pulses of finite length can be expressed by a successive application of equation (7.79) where the initial positions for each step are the final positions from the preceding step. An example is shown in figure 7.6.

8 Numerical Calculations

The applicability of the approximations leading to the analytical result in equations (7.56), (7.72), (7.74), and (7.79) are tested by micromagnetic simulations. In this section we will discuss the

results of micromagnetic simulations for magnetic thin-film elements of different sizes containing vortices and antivortices with different polarizations and chiralities.

The results are compared with the analytical predictions. The analytical calculations yield the position of the vortex core. In the numerical simulations the position of the vortex is defined by the maximum amplitude of the out-of-plane magnetization. To determine this maximum, the simulation cell with maximum out-of-plane magnetization and its next neighbors are interpolated with a polynomial of second order (see appendix C for details). For the simulations the material parameters of permalloy are used, that are an exchange constant of A = 13·10⁻¹² J/m and a saturation magnetization of M_s = 8·10⁵ A/m. Numerical calculations are performed for vortices in square film elements with edge lengths l from 200 nm to 1000 nm and film thicknesses dfrom 10 nm to 30 nm. The magnetization dynamics is investigated mainly for square shaped magnetic film elements with length l = 200 nm and thickness d= 20 nm and with length l= 500 nm and thicknessd= 10 nm. This system sizes allow for reasonable computing time. The results that are presented in this section have been published in references [157, 178, 179].

8.1 Calculation of the Groundstate

For the simulations the magnetization configuration of a vortex in the groundstate for all lateral sizes l and thicknesses dis needed. As one can see from equation (7.2) the Gilbert damping parameter α occurs only in a term that is proportional to the time derivative of the magnetization. When the simulation reaches its groundstate this time derivative becomes zero. Thus the groundstate is not affected by the choice of the parameter α. For calculations of the groundstate it is convenient to use a high Gilbert damping that speeds up the calculation since there is a faster dissipation of energy. Here a value ofα = 0.5 is used. Since the LLG equation provides only the time derivative of the magnetization it is inevitable to supply an initial magnetization for the simulations. The final state depends strongly on this initial magnetization. For instance the polarization and chirality of the final state are affected by the initial magnetization. There are even initial magnetizations that do not relax to a vortex state. In the calculations that are presented here the groundstate for l= 200 nm andd= 20 nm is calculated using the initial magnetization that is given by the pattern

M~ Ms

=















(0,0, p), x²+y²≤100 nm²

(0, c,0), x²+y²>100 nm²,y ≤x, and y >−x (c,0,0), x²+y²>100 nm²,y ≤x, and y≤ −x (0,−c,0), x²+y²>100 nm²,y > x, and y≤ −x (−c,0,0), x²+y²>100 nm²,y > x, and y >−x

(8.1)

that is depicted in figure 8.1 for polarization p= 1 and chirality c= 1. Subsequently, the ground-state obtained for l = 200 nm and d = 20 nm is scaled and serves as an initial state for the remaining samples.

Im Dokument Current-Driven Magnetization Dynamics : Analytical Modeling and Numerical Simulation (Seite 42-50)